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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sibfima | Structured version Visualization version GIF version |
Description: Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.) |
Ref | Expression |
---|---|
sitgval.b | β’ π΅ = (Baseβπ) |
sitgval.j | β’ π½ = (TopOpenβπ) |
sitgval.s | β’ π = (sigaGenβπ½) |
sitgval.0 | β’ 0 = (0gβπ) |
sitgval.x | β’ Β· = ( Β·π βπ) |
sitgval.h | β’ π» = (βHomβ(Scalarβπ)) |
sitgval.1 | β’ (π β π β π) |
sitgval.2 | β’ (π β π β βͺ ran measures) |
sibfmbl.1 | β’ (π β πΉ β dom (πsitgπ)) |
Ref | Expression |
---|---|
sibfima | β’ ((π β§ π΄ β (ran πΉ β { 0 })) β (πβ(β‘πΉ β {π΄})) β (0[,)+β)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sibfmbl.1 | . . . 4 β’ (π β πΉ β dom (πsitgπ)) | |
2 | sitgval.b | . . . . 5 β’ π΅ = (Baseβπ) | |
3 | sitgval.j | . . . . 5 β’ π½ = (TopOpenβπ) | |
4 | sitgval.s | . . . . 5 β’ π = (sigaGenβπ½) | |
5 | sitgval.0 | . . . . 5 β’ 0 = (0gβπ) | |
6 | sitgval.x | . . . . 5 β’ Β· = ( Β·π βπ) | |
7 | sitgval.h | . . . . 5 β’ π» = (βHomβ(Scalarβπ)) | |
8 | sitgval.1 | . . . . 5 β’ (π β π β π) | |
9 | sitgval.2 | . . . . 5 β’ (π β π β βͺ ran measures) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | issibf 33630 | . . . 4 β’ (π β (πΉ β dom (πsitgπ) β (πΉ β (dom πMblFnMπ) β§ ran πΉ β Fin β§ βπ₯ β (ran πΉ β { 0 })(πβ(β‘πΉ β {π₯})) β (0[,)+β)))) |
11 | 1, 10 | mpbid 231 | . . 3 β’ (π β (πΉ β (dom πMblFnMπ) β§ ran πΉ β Fin β§ βπ₯ β (ran πΉ β { 0 })(πβ(β‘πΉ β {π₯})) β (0[,)+β))) |
12 | 11 | simp3d 1142 | . 2 β’ (π β βπ₯ β (ran πΉ β { 0 })(πβ(β‘πΉ β {π₯})) β (0[,)+β)) |
13 | sneq 4637 | . . . . . 6 β’ (π₯ = π΄ β {π₯} = {π΄}) | |
14 | 13 | imaeq2d 6058 | . . . . 5 β’ (π₯ = π΄ β (β‘πΉ β {π₯}) = (β‘πΉ β {π΄})) |
15 | 14 | fveq2d 6894 | . . . 4 β’ (π₯ = π΄ β (πβ(β‘πΉ β {π₯})) = (πβ(β‘πΉ β {π΄}))) |
16 | 15 | eleq1d 2816 | . . 3 β’ (π₯ = π΄ β ((πβ(β‘πΉ β {π₯})) β (0[,)+β) β (πβ(β‘πΉ β {π΄})) β (0[,)+β))) |
17 | 16 | rspcv 3607 | . 2 β’ (π΄ β (ran πΉ β { 0 }) β (βπ₯ β (ran πΉ β { 0 })(πβ(β‘πΉ β {π₯})) β (0[,)+β) β (πβ(β‘πΉ β {π΄})) β (0[,)+β))) |
18 | 12, 17 | mpan9 505 | 1 β’ ((π β§ π΄ β (ran πΉ β { 0 })) β (πβ(β‘πΉ β {π΄})) β (0[,)+β)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1085 = wceq 1539 β wcel 2104 βwral 3059 β cdif 3944 {csn 4627 βͺ cuni 4907 β‘ccnv 5674 dom cdm 5675 ran crn 5676 β cima 5678 βcfv 6542 (class class class)co 7411 Fincfn 8941 0cc0 11112 +βcpnf 11249 [,)cico 13330 Basecbs 17148 Scalarcsca 17204 Β·π cvsca 17205 TopOpenctopn 17371 0gc0g 17389 βHomcrrh 33271 sigaGencsigagen 33434 measurescmeas 33491 MblFnMcmbfm 33545 sitgcsitg 33626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-sitg 33627 |
This theorem is referenced by: sibfinima 33636 sitgfval 33638 sitgclg 33639 |
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